3.165 \(\int \frac{\sinh ^7(x)}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=140 \[ \frac{\left (a^2-3 b^2\right ) \cosh ^4(x)}{4 b^3}-\frac{a \left (a^2-3 b^2\right ) \cosh ^3(x)}{3 b^4}+\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \cosh ^2(x)}{2 b^5}-\frac{a \left (-3 a^2 b^2+a^4+3 b^4\right ) \cosh (x)}{b^6}+\frac{\left (a^2-b^2\right )^3 \log (a+b \cosh (x))}{b^7}-\frac{a \cosh ^5(x)}{5 b^2}+\frac{\cosh ^6(x)}{6 b} \]

[Out]

-((a*(a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x])/b^6) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x]^2)/(2*b^5) - (a*(a^2 - 3*b^2
)*Cosh[x]^3)/(3*b^4) + ((a^2 - 3*b^2)*Cosh[x]^4)/(4*b^3) - (a*Cosh[x]^5)/(5*b^2) + Cosh[x]^6/(6*b) + ((a^2 - b
^2)^3*Log[a + b*Cosh[x]])/b^7

________________________________________________________________________________________

Rubi [A]  time = 0.167012, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac{\left (a^2-3 b^2\right ) \cosh ^4(x)}{4 b^3}-\frac{a \left (a^2-3 b^2\right ) \cosh ^3(x)}{3 b^4}+\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \cosh ^2(x)}{2 b^5}-\frac{a \left (-3 a^2 b^2+a^4+3 b^4\right ) \cosh (x)}{b^6}+\frac{\left (a^2-b^2\right )^3 \log (a+b \cosh (x))}{b^7}-\frac{a \cosh ^5(x)}{5 b^2}+\frac{\cosh ^6(x)}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^7/(a + b*Cosh[x]),x]

[Out]

-((a*(a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x])/b^6) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x]^2)/(2*b^5) - (a*(a^2 - 3*b^2
)*Cosh[x]^3)/(3*b^4) + ((a^2 - 3*b^2)*Cosh[x]^4)/(4*b^3) - (a*Cosh[x]^5)/(5*b^2) + Cosh[x]^6/(6*b) + ((a^2 - b
^2)^3*Log[a + b*Cosh[x]])/b^7

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\sinh ^7(x)}{a+b \cosh (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^3}{a+x} \, dx,x,b \cosh (x)\right )}{b^7}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a^5 \left (1+\frac{3 b^2 \left (-a^2+b^2\right )}{a^4}\right )-\left (a^4-3 a^2 b^2+3 b^4\right ) x+a \left (a^2-3 b^2\right ) x^2-\left (a^2-3 b^2\right ) x^3+a x^4-x^5-\frac{\left (a^2-b^2\right )^3}{a+x}\right ) \, dx,x,b \cosh (x)\right )}{b^7}\\ &=-\frac{a \left (a^4-3 a^2 b^2+3 b^4\right ) \cosh (x)}{b^6}+\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \cosh ^2(x)}{2 b^5}-\frac{a \left (a^2-3 b^2\right ) \cosh ^3(x)}{3 b^4}+\frac{\left (a^2-3 b^2\right ) \cosh ^4(x)}{4 b^3}-\frac{a \cosh ^5(x)}{5 b^2}+\frac{\cosh ^6(x)}{6 b}+\frac{\left (a^2-b^2\right )^3 \log (a+b \cosh (x))}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.17459, size = 144, normalized size = 1.03 \[ \frac{-30 b^4 \left (2 b^2-a^2\right ) \cosh (4 x)+15 b^2 \left (-40 a^2 b^2+16 a^4+29 b^4\right ) \cosh (2 x)-120 a b \left (-22 a^2 b^2+8 a^4+19 b^4\right ) \cosh (x)+960 \left (a^2-b^2\right )^3 \log (a+b \cosh (x))-12 a b^5 \cosh (5 x)-20 a b^3 (2 a-3 b) (2 a+3 b) \cosh (3 x)+5 b^6 \cosh (6 x)}{960 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^7/(a + b*Cosh[x]),x]

[Out]

(-120*a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cosh[x] + 15*b^2*(16*a^4 - 40*a^2*b^2 + 29*b^4)*Cosh[2*x] - 20*a*(2*a
- 3*b)*b^3*(2*a + 3*b)*Cosh[3*x] - 30*b^4*(-a^2 + 2*b^2)*Cosh[4*x] - 12*a*b^5*Cosh[5*x] + 5*b^6*Cosh[6*x] + 96
0*(a^2 - b^2)^3*Log[a + b*Cosh[x]])/(960*b^7)

________________________________________________________________________________________

Maple [B]  time = 0.034, size = 1039, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^7/(a+b*cosh(x)),x)

[Out]

7/12/b/(tanh(1/2*x)+1)^3+5/16/b/(tanh(1/2*x)+1)^2-11/16/b/(tanh(1/2*x)+1)-7/12/b/(tanh(1/2*x)-1)^3+5/16/b/(tan
h(1/2*x)-1)^2+11/16/b/(tanh(1/2*x)-1)+1/b*ln(tanh(1/2*x)+1)+1/b*ln(tanh(1/2*x)-1)-3/b^2/(a-b)*ln(a*tanh(1/2*x)
^2-tanh(1/2*x)^2*b-a-b)*a^2-1/b/(a-b)*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)*a+1/b^7/(a-b)*ln(a*tanh(1/2*x)^2
-tanh(1/2*x)^2*b-a-b)*a^7-1/b^6/(a-b)*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)*a^6-3/b^5/(a-b)*ln(a*tanh(1/2*x)
^2-tanh(1/2*x)^2*b-a-b)*a^5+3/b^4/(a-b)*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)*a^4+3/b^3/(a-b)*ln(a*tanh(1/2*
x)^2-tanh(1/2*x)^2*b-a-b)*a^3+1/2/b/(tanh(1/2*x)-1)^5+1/8/b/(tanh(1/2*x)-1)^4+1/6/b/(tanh(1/2*x)+1)^6+1/6/b/(t
anh(1/2*x)-1)^6+1/(a-b)*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)-1/2/b/(tanh(1/2*x)+1)^5+1/8/b/(tanh(1/2*x)+1)^
4-3/b^3*ln(tanh(1/2*x)+1)*a^2-3/b^3*ln(tanh(1/2*x)-1)*a^2-7/8/b^2/(tanh(1/2*x)+1)^2*a+9/8/b^3/(tanh(1/2*x)+1)*
a^2-15/8/b^2/(tanh(1/2*x)+1)*a-7/8/b^2/(tanh(1/2*x)-1)^2*a-9/8/b^3/(tanh(1/2*x)-1)*a^2+15/8/b^2/(tanh(1/2*x)-1
)*a-1/5/b^2/(tanh(1/2*x)+1)^5*a+1/4/b^3/(tanh(1/2*x)+1)^4*a^2+1/2/b^2/(tanh(1/2*x)+1)^4*a-1/3/b^4/(tanh(1/2*x)
+1)^3*a^3-1/2/b^3/(tanh(1/2*x)+1)^3*a^2+1/4/b^2/(tanh(1/2*x)+1)^3*a+3/b^5*ln(tanh(1/2*x)+1)*a^4+1/2/b^5/(tanh(
1/2*x)+1)^2*a^4+1/2/b^4/(tanh(1/2*x)+1)^2*a^3-7/8/b^3/(tanh(1/2*x)+1)^2*a^2-1/b^6/(tanh(1/2*x)+1)*a^5-1/2/b^5/
(tanh(1/2*x)+1)*a^4+5/2/b^4/(tanh(1/2*x)+1)*a^3+1/5/b^2/(tanh(1/2*x)-1)^5*a+1/4/b^3/(tanh(1/2*x)-1)^4*a^2+1/2/
b^2/(tanh(1/2*x)-1)^4*a+1/3/b^4/(tanh(1/2*x)-1)^3*a^3+1/2/b^3/(tanh(1/2*x)-1)^3*a^2-1/4/b^2/(tanh(1/2*x)-1)^3*
a-1/b^7*ln(tanh(1/2*x)-1)*a^6+3/b^5*ln(tanh(1/2*x)-1)*a^4+1/2/b^5/(tanh(1/2*x)-1)^2*a^4+1/2/b^4/(tanh(1/2*x)-1
)^2*a^3-7/8/b^3/(tanh(1/2*x)-1)^2*a^2+1/b^6/(tanh(1/2*x)-1)*a^5+1/2/b^5/(tanh(1/2*x)-1)*a^4-5/2/b^4/(tanh(1/2*
x)-1)*a^3-1/b^7*ln(tanh(1/2*x)+1)*a^6

________________________________________________________________________________________

Maxima [B]  time = 1.06457, size = 419, normalized size = 2.99 \begin{align*} -\frac{{\left (12 \, a b^{4} e^{\left (-x\right )} - 5 \, b^{5} - 30 \,{\left (a^{2} b^{3} - 2 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \,{\left (4 \, a^{3} b^{2} - 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} - 15 \,{\left (16 \, a^{4} b - 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-4 \, x\right )} + 120 \,{\left (8 \, a^{5} - 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-5 \, x\right )}\right )} e^{\left (6 \, x\right )}}{1920 \, b^{6}} - \frac{12 \, a b^{4} e^{\left (-5 \, x\right )} - 5 \, b^{5} e^{\left (-6 \, x\right )} + 120 \,{\left (8 \, a^{5} - 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-x\right )} - 15 \,{\left (16 \, a^{4} b - 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \,{\left (4 \, a^{3} b^{2} - 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} - 30 \,{\left (a^{2} b^{3} - 2 \, b^{5}\right )} e^{\left (-4 \, x\right )}}{1920 \, b^{6}} + \frac{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} x}{b^{7}} + \frac{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^7/(a+b*cosh(x)),x, algorithm="maxima")

[Out]

-1/1920*(12*a*b^4*e^(-x) - 5*b^5 - 30*(a^2*b^3 - 2*b^5)*e^(-2*x) + 20*(4*a^3*b^2 - 9*a*b^4)*e^(-3*x) - 15*(16*
a^4*b - 40*a^2*b^3 + 29*b^5)*e^(-4*x) + 120*(8*a^5 - 22*a^3*b^2 + 19*a*b^4)*e^(-5*x))*e^(6*x)/b^6 - 1/1920*(12
*a*b^4*e^(-5*x) - 5*b^5*e^(-6*x) + 120*(8*a^5 - 22*a^3*b^2 + 19*a*b^4)*e^(-x) - 15*(16*a^4*b - 40*a^2*b^3 + 29
*b^5)*e^(-2*x) + 20*(4*a^3*b^2 - 9*a*b^4)*e^(-3*x) - 30*(a^2*b^3 - 2*b^5)*e^(-4*x))/b^6 + (a^6 - 3*a^4*b^2 + 3
*a^2*b^4 - b^6)*x/b^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(2*a*e^(-x) + b*e^(-2*x) + b)/b^7

________________________________________________________________________________________

Fricas [B]  time = 2.15659, size = 5296, normalized size = 37.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^7/(a+b*cosh(x)),x, algorithm="fricas")

[Out]

1/1920*(5*b^6*cosh(x)^12 + 5*b^6*sinh(x)^12 - 12*a*b^5*cosh(x)^11 + 12*(5*b^6*cosh(x) - a*b^5)*sinh(x)^11 + 30
*(a^2*b^4 - 2*b^6)*cosh(x)^10 + 6*(55*b^6*cosh(x)^2 - 22*a*b^5*cosh(x) + 5*a^2*b^4 - 10*b^6)*sinh(x)^10 - 20*(
4*a^3*b^3 - 9*a*b^5)*cosh(x)^9 + 20*(55*b^6*cosh(x)^3 - 33*a*b^5*cosh(x)^2 - 4*a^3*b^3 + 9*a*b^5 + 15*(a^2*b^4
- 2*b^6)*cosh(x))*sinh(x)^9 + 15*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^8 + 15*(165*b^6*cosh(x)^4 - 132*a
*b^5*cosh(x)^3 + 16*a^4*b^2 - 40*a^2*b^4 + 29*b^6 + 90*(a^2*b^4 - 2*b^6)*cosh(x)^2 - 12*(4*a^3*b^3 - 9*a*b^5)*
cosh(x))*sinh(x)^8 - 1920*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^6 - 120*(8*a^5*b - 22*a^3*b^3 + 19*a*b
^5)*cosh(x)^7 + 120*(33*b^6*cosh(x)^5 - 33*a*b^5*cosh(x)^4 - 8*a^5*b + 22*a^3*b^3 - 19*a*b^5 + 30*(a^2*b^4 - 2
*b^6)*cosh(x)^3 - 6*(4*a^3*b^3 - 9*a*b^5)*cosh(x)^2 + (16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x))*sinh(x)^7 -
12*a*b^5*cosh(x) + 12*(385*b^6*cosh(x)^6 - 462*a*b^5*cosh(x)^5 + 525*(a^2*b^4 - 2*b^6)*cosh(x)^4 - 140*(4*a^3*
b^3 - 9*a*b^5)*cosh(x)^3 + 35*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^2 - 160*(a^6 - 3*a^4*b^2 + 3*a^2*b^4
- b^6)*x - 70*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x))*sinh(x)^6 + 5*b^6 - 120*(8*a^5*b - 22*a^3*b^3 + 19*a*
b^5)*cosh(x)^5 + 24*(165*b^6*cosh(x)^7 - 231*a*b^5*cosh(x)^6 - 40*a^5*b + 110*a^3*b^3 - 95*a*b^5 + 315*(a^2*b^
4 - 2*b^6)*cosh(x)^5 - 105*(4*a^3*b^3 - 9*a*b^5)*cosh(x)^4 + 35*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^3 -
480*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x) - 105*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x)^2)*sinh(x)^
5 + 15*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^4 + 15*(165*b^6*cosh(x)^8 - 264*a*b^5*cosh(x)^7 + 420*(a^2*b
^4 - 2*b^6)*cosh(x)^6 + 16*a^4*b^2 - 40*a^2*b^4 + 29*b^6 - 168*(4*a^3*b^3 - 9*a*b^5)*cosh(x)^5 + 70*(16*a^4*b^
2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^4 - 1920*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^2 - 280*(8*a^5*b - 22*
a^3*b^3 + 19*a*b^5)*cosh(x)^3 - 40*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x))*sinh(x)^4 - 20*(4*a^3*b^3 - 9*a*
b^5)*cosh(x)^3 + 20*(55*b^6*cosh(x)^9 - 99*a*b^5*cosh(x)^8 + 180*(a^2*b^4 - 2*b^6)*cosh(x)^7 - 84*(4*a^3*b^3 -
9*a*b^5)*cosh(x)^6 - 4*a^3*b^3 + 9*a*b^5 + 42*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^5 - 1920*(a^6 - 3*a^
4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^3 - 210*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x)^4 - 60*(8*a^5*b - 22*a^3*
b^3 + 19*a*b^5)*cosh(x)^2 + 3*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x))*sinh(x)^3 + 30*(a^2*b^4 - 2*b^6)*cos
h(x)^2 + 30*(11*b^6*cosh(x)^10 - 22*a*b^5*cosh(x)^9 + 45*(a^2*b^4 - 2*b^6)*cosh(x)^8 - 24*(4*a^3*b^3 - 9*a*b^5
)*cosh(x)^7 + 14*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^6 + a^2*b^4 - 2*b^6 - 960*(a^6 - 3*a^4*b^2 + 3*a^2
*b^4 - b^6)*x*cosh(x)^4 - 84*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x)^5 - 40*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5
)*cosh(x)^3 + 3*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^2 - 2*(4*a^3*b^3 - 9*a*b^5)*cosh(x))*sinh(x)^2 + 19
20*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5*sinh(x) +
15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4*sinh(x)^2 + 20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3*
sinh(x)^3 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2*sinh(x)^4 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)
*cosh(x)*sinh(x)^5 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^6)*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x)))
+ 12*(5*b^6*cosh(x)^11 - 11*a*b^5*cosh(x)^10 + 25*(a^2*b^4 - 2*b^6)*cosh(x)^9 - 15*(4*a^3*b^3 - 9*a*b^5)*cosh
(x)^8 + 10*(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^7 - 960*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^5
- 70*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x)^6 - a*b^5 - 50*(8*a^5*b - 22*a^3*b^3 + 19*a*b^5)*cosh(x)^4 + 5*
(16*a^4*b^2 - 40*a^2*b^4 + 29*b^6)*cosh(x)^3 - 5*(4*a^3*b^3 - 9*a*b^5)*cosh(x)^2 + 5*(a^2*b^4 - 2*b^6)*cosh(x)
)*sinh(x))/(b^7*cosh(x)^6 + 6*b^7*cosh(x)^5*sinh(x) + 15*b^7*cosh(x)^4*sinh(x)^2 + 20*b^7*cosh(x)^3*sinh(x)^3
+ 15*b^7*cosh(x)^2*sinh(x)^4 + 6*b^7*cosh(x)*sinh(x)^5 + b^7*sinh(x)^6)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**7/(a+b*cosh(x)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.13178, size = 309, normalized size = 2.21 \begin{align*} \frac{5 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{6} - 12 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}^{5} + 30 \, a^{2} b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 90 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 80 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 240 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 240 \, a^{4} b{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 720 \, a^{2} b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 720 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 960 \, a^{5}{\left (e^{\left (-x\right )} + e^{x}\right )} + 2880 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )} - 2880 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}}{1920 \, b^{6}} + \frac{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^7/(a+b*cosh(x)),x, algorithm="giac")

[Out]

1/1920*(5*b^5*(e^(-x) + e^x)^6 - 12*a*b^4*(e^(-x) + e^x)^5 + 30*a^2*b^3*(e^(-x) + e^x)^4 - 90*b^5*(e^(-x) + e^
x)^4 - 80*a^3*b^2*(e^(-x) + e^x)^3 + 240*a*b^4*(e^(-x) + e^x)^3 + 240*a^4*b*(e^(-x) + e^x)^2 - 720*a^2*b^3*(e^
(-x) + e^x)^2 + 720*b^5*(e^(-x) + e^x)^2 - 960*a^5*(e^(-x) + e^x) + 2880*a^3*b^2*(e^(-x) + e^x) - 2880*a*b^4*(
e^(-x) + e^x))/b^6 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(abs(b*(e^(-x) + e^x) + 2*a))/b^7